Log-normal distribution

Log-normal

Probability density function
Cumulative distribution function Distribution function of the log-normal distribution (with μ = 0 )
Notation	$\ln {\mathcal {N}}(\mu ,\,\sigma ^{2})$
Parameters	σ2 > 0 — squared scale (real), μ ∈ R — location
Support	x ∈ (0, +∞)
PDF	${\frac {1}{x{\sqrt {2\pi \sigma ^{2}}}}}\,e^{-{\frac {\left(\ln x-\mu \right)^{2}}{2\sigma ^{2}}}}$
CDF	${\frac {1}{2}}+{\frac {1}{2}}\,\mathrm {erf} {\Big [}{\frac {\ln x-\mu }{\sqrt {2\sigma ^{2}}}}{\Big ]}$
Mean	$e^{\mu +\sigma ^{2}/2}$
Median	$e^{\mu }\,$
Mode	$e^{\mu -\sigma ^{2}}$
Variance	$(e^{\sigma ^{2}}\!\!-1)e^{2\mu +\sigma ^{2}}$
Skewness	$(e^{\sigma ^{2}}\!\!+2){\sqrt {e^{\sigma ^{2}}\!\!-1}}$
Excess kurtosis	$e^{4\sigma ^{2}}\!\!+2e^{3\sigma ^{2}}\!\!+3e^{2\sigma ^{2}}\!\!-6$
Entropy	${\frac {1}{2}}+{\frac {1}{2}}\ln(2\pi \sigma ^{2})+\mu$
MGF	(defined only on the negative half-axis, see text)
CF	representation $\sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}$ is asymptotically divergent but sufficient for numerical purposes
Fisher information	${\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}$

In probability theory, a log-normal distribution is a probability distribution of a random variable whose logarithm is normally distributed. If X is a random variable with a normal distribution, then Y = exp(X) has a log-normal distribution; likewise, if Y is log-normally distributed, then X = log(Y) is normally distributed. (This is true regardless of the base of the logarithmic function: if log_a(Y) is normally distributed, then so is log_b(Y), for any two positive numbers a, b ≠ 1.)

Log-normal is also written log normal or lognormal. It is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton.

A variable might be modeled as log-normal if it can be thought of as the multiplicative product of many independent random variables each of which is positive. For example, in finance, a long-term discount factor can be derived from the product of short-term discount factors. In wireless communication, the attenuation caused by shadowing or slow fading from random objects is often assumed to be log-normally distributed. See log-distance path loss model.

Characterization

Probability density function

The probability density function of a log-normal distribution is:

$f_{X}(x;\mu ,\sigma )={\frac {1}{x\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(\ln x-\mu )^{2}}{2\sigma ^{2}}}},\ \ x>0$

where μ and σ are the mean and standard deviation of the variable’s natural logarithm (by definition, the variable’s logarithm is normally distributed).

Cumulative distribution function

$F_{X}(x;\mu ,\sigma )={\frac {1}{2}}\operatorname {erfc} \!\left[-{\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right]=\Phi {\bigg (}{\frac {\ln x-\mu }{\sigma }}{\bigg )},$

where erfc is the complementary error function, and Φ is the standard normal cdf.

Mean and standard deviation

If X is a lognormally distributed variable, its expected value (which can be assumed to represent the arithmetic mean), variance, and standard deviation are

${\begin{aligned}&\mathrm {E} [X]=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}},\\&\mathrm {Var} [X]=(e^{\sigma ^{2}}-1)e^{2\mu +\sigma ^{2}}\\&\mathrm {s.d} [X]={\sqrt {\mathrm {Var} [X]}}=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}{\sqrt {e^{\sigma ^{2}}-1}}\end{aligned}}$

Equivalently, parameters μ and σ can be obtained if the expected value and variance are known:

${\begin{aligned}\mu &=\ln(\mathrm {E} [X])-{\frac {1}{2}}\ln \!\left(1+{\frac {\mathrm {Var} [X]}{\mathrm {E} [X]^{2}}}\right),\\\sigma ^{2}&=\ln \!\left(1+{\frac {\mathrm {Var} [X]}{\mathrm {E} [X]^{2}}}\right).\end{aligned}}$

The geometric mean of the log-normal distribution is $e^{\mu }$, and the geometric standard deviation is equal to $e^{\sigma }$. Subsequently, the geometric mean (m_g) can alternatively be estimated from an arithmetic mean (m_a) by:

$m_{g}=m_{a}e^{-{\tfrac {1}{2}}\sigma ^{2}}$

Mode and median

The mode is the point of global maximum of the pdf function. In particular, it solves the equation (ln ƒ)′ = 0:

$\mathrm {Mode} [X]=e^{\mu -\sigma ^{2}}.$

The median is such a point where F_X = ½:

$\mathrm {Med} [X]=e^{\mu }\,.$

Confidence interval

If X is distributed log-normally with parameters μ and σ, then the (1 − α)-confidence interval for X will be

$X\in {\Big [}\,e^{\mu -\sigma q^{*}}\!,\ e^{\mu +\sigma q^{*}}\,{\Big ]},$

where q* is the (1 − α/2)-quantile of the standard normal distribution: q* = Φ⁻¹(1 − α/2).

Moments

For any real or complex number s, the s^th moment of log-normal X is given by

$\operatorname {E} [X^{s}]=e^{s\mu +{\tfrac {1}{2}}s^{2}\sigma ^{2}}.$

A log-normal distribution is not uniquely determined by its moments E[X^k] for k ≥ 1, that is, there exists some other distribution with the same moments for all k. In fact, there is a whole family of distributions with the same moments as the log-normal distribution.

Characteristic function and moment generating function

The characteristic function E[e ^itX] has a number of representations. The integral itself converges for Im(t) ≤ 0. The simplest representation is obtained by Taylor expanding e ^itX and using formula for moments above.

$\varphi (t)=\sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}.$

This series representation is divergent for Re(σ²) > 0, however it is sufficient for numerically evaluating the characteristic function at positive $\sigma$ as long as the upper limit in sum above is kept bounded, n ≤ N, where

$\max(|t|,|\mu |)\ll N\ll {\frac {2}{\sigma ^{2}}}\ln {\frac {2}{\sigma ^{2}}}$

and σ² < 0.1. To bring the numerical values of parameters μ, σ into the domain where strong inequality holds true one could use the fact that if X is log-normally distributed then X^m is also log-normally distributed with parameters μm, σm. Since $\mu \sigma ^{2}\propto m^{3}$, the inequality could be satisfied for sufficiently small m. The sum of series first converges to the value of φ(t) with arbitrary high accuracy if m is small enough, and left part of the strong inequality is satisfied. If considerably larger number of terms are taken into account the sum eventually diverges when the right part of the strong inequality is no longer valid.

Another useful representation was derived by Roy Lepnik (see references by this author and by Daniel Dufresne below) by means of double Taylor expansion of e^{(ln x − μ)2/(2σ2)}.

The moment-generating function for the log-normal distribution does not exist on the domain R, but only exists on the half-interval (−∞, 0].

Partial expectation

The partial expectation of a random variable X with respect to a threshold k is defined as g(k) = E[X | X > k]P[X > k]. For a log-normal random variable the partial expectation is given by

$g(k)=\int _{k}^{\infty }\!xf(x)\,dx=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}\,\Phi \!\left({\frac {\mu +\sigma ^{2}-\ln k}{\sigma }}\right).$

This formula has applications in insurance and economics, it is used in solving the partial differential equation leading to the Black–Scholes formula.

Curve shape

A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).

Occurrence

• In biology, variables whose logarithms tend to have a normal distribution include:
  • Measures of size of living tissue (length, height, skin area, weight);
  • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth;
  • Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations).

Subsequently, reference ranges for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.

• In finance, in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as Benoît Mandelbrot have argued that log-Levy distributions which possesses heavy tails would be a more appropriate model, in particular for the analysis for stock market crashes.

Maximum likelihood estimation of parameters

For determining the maximum likelihood estimators of the log-normal distribution parameters μ and σ, we can use the same procedure as for the normal distribution. To avoid repetition, we observe that

$f_{L}(x;\mu ,\sigma )=\prod _{i=1}^{n}\left({\frac {1}{x}}_{i}\right)\,f_{N}(\ln x;\mu ,\sigma )$

where by ƒ_L we denote the probability density function of the log-normal distribution and by ƒ_N that of the normal distribution. Therefore, using the same indices to denote distributions, we can write the log-likelihood function thus:

${\begin{aligned}\ell _{L}(\mu ,\sigma |x_{1},x_{2},\dots ,x_{n})&{}=-\sum _{k}\ln x_{k}+\ell _{N}(\mu ,\sigma |\ln x_{1},\ln x_{2},\dots ,\ln x_{n})\\&{}=\operatorname {constant} +\ell _{N}(\mu ,\sigma |\ln x_{1},\ln x_{2},\dots ,\ln x_{n}).\end{aligned}}$

Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, ℓ_L ℓ_L and ℓ_N, reach their maximum with the same μ and σ. Hence, using the formulas for the normal distribution maximum likelihood parameter estimators and the equality above, we deduce that for the log-normal distribution it holds that

${\widehat {\mu }}={\frac {\sum _{k}\ln x_{k}}{n}},\ {\widehat {\sigma }}^{2}={\frac {\sum _{k}\left(\ln x_{k}-{\widehat {\mu }}\right)^{2}}{n}}.$

Generating log-normally-distributed random variates

Given a random variate N drawn from the normal distribution with 0 mean and 1 standard deviation, then the variate

$X=e^{\mu +\sigma N}\,$

has a Log-normal distribution with parameters $\mu$ and $\sigma$.

Related distributions

• If $X\sim {\mathcal {N}}(\mu ,\sigma ^{2})$ is a normal distribution, then $\exp(X)\sim \operatorname {Log-{\mathcal {N}}} (\mu ,\sigma ^{2}).$
• If $X\sim \operatorname {Log-{\mathcal {N}}} (\mu ,\sigma ^{2})$ is distributed log-normally, then $\ln(X)\sim {\mathcal {N}}(\mu ,\sigma ^{2})$ is a normal random variable.
• If $X_{j}\sim \operatorname {Log-{\mathcal {N}}} (\mu _{j},\sigma _{j}^{2})$ are n independent log-normally distributed variables, and $Y=\textstyle \prod _{j=1}^{n}X_{j}$, then Y is also distributed log-normally:

  $Y\sim \operatorname {Log-{\mathcal {N}}} {\Big (}\textstyle \sum _{j=1}^{n}\mu _{j},\ \sum _{j=1}^{n}\sigma _{j}^{2}{\Big )}.$

• Let $X_{j}\sim \operatorname {Log-{\mathcal {N}}} (\mu _{j},\sigma _{j}^{2})\$ be independent log-normally distributed variables with possibly varying σ and μ parameters, and $Y=\textstyle \sum _{j=1}^{n}X_{j}$. The distribution of Y has no closed-form expression, but can be reasonably approximated by another log-normal distribution Z at the right tail. Its probability density function at the neighborhood of 0 is characterized in (Gao et al., 2009) and it does not resemble any log-normal distribution. A commonly used approximation (due to Fenton and Wilkinson) is obtained by matching the mean and variance:

  ${\begin{aligned}\sigma _{Z}^{2}&=\log \!\left[{\frac {\sum e^{2\mu _{j}+\sigma _{j}^{2}}(e^{\sigma _{j}^{2}}-1)}{(\sum e^{\mu _{j}+\sigma _{j}^{2}/2})^{2}}}+1\right],\\\mu _{Z}&=\log \!\left[\sum e^{\mu _{j}+\sigma _{j}^{2}/2}\right]-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}$

  In the case that all $X_{j}$ have the same variance parameter $\sigma _{j}=\sigma$, these formulas simplify to

  ${\begin{aligned}\sigma _{Z}^{2}&=\log \!\left[(e^{\sigma ^{2}}-1){\frac {\sum e^{2\mu _{j}}}{(\sum e^{\mu _{j}})^{2}}}+1\right],\\\mu _{Z}&=\log \!\left[\sum e^{\mu _{j}}\right]+{\frac {\sigma ^{2}}{2}}-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}$

• If $X\sim \operatorname {Log-{\mathcal {N}}} (\mu ,\sigma ^{2})$, then X + c is said to have a shifted log-normal distribution with support x ∈ (c, +∞). E[X + c] = E[X] + c, Var[X + c] = Var[X].
• If $X\sim \operatorname {Log-{\mathcal {N}}} (\mu ,\sigma ^{2})$, then Y = aX is also log-normal, $Y\sim \operatorname {Log-{\mathcal {N}}} (\ln a+\mu ,\ \sigma ^{2}).$
• If $X\sim \operatorname {Log-{\mathcal {N}}} (\mu ,\sigma ^{2})$, then Y = 1⁄X is also log-normal, $Y\sim \operatorname {Log-{\mathcal {N}}} (-\mu ,\ \sigma ^{2}).$
• If $X\sim \operatorname {Log-{\mathcal {N}}} (\mu ,\sigma ^{2})$ and a ≠ 0, then Y = X^a is also log-normal, $Y\sim \operatorname {Log-{\mathcal {N}}} (a\mu ,\ a^{2}\sigma ^{2}).$

Similar distributions

• A substitute for the log-normal whose integral can be expressed in terms of more elementary functions (Swamee, 2002) can be obtained based on the logistic distribution to get the CDF

$F(x;\mu ,\sigma )=\left[\left({\frac {e^{\mu }}{x}}\right)^{\pi /(\sigma {\sqrt {3}})}+1\right]^{-1}.$

This is a log-logistic distribution.

Further reading

• Robert Brooks, Jon Corson, and J. Donal Wales. "The Pricing of Index Options When the Underlying Assets All Follow a Lognormal Diffusion", in Advances in Futures and Options Research, volume 7, 1994.

Citations

• The Lognormal Distribution, Aitchison, J. and Brown, J.A.C. (1957)
• Log-normal Distributions across the Sciences: Keys and Clues, E. Limpert, W. Stahel and M. Abbt,. BioScience, 51 (5), p. 341–352 (2001).
• Eric W. Weisstein et al. Log Normal Distribution at MathWorld. Electronic document, retrieved October 26, 2006.
• Swamee, P.K. (2002). Near Lognormal Distribution, Journal of Hydrologic Engineering. 7(6): 441-444
• Roy B. Leipnik (1991), On Lognormal Random Variables: I - The Characteristic Function, Journal of the Australian Mathematical Society Series B, vol. 32, pp 327–347.
• Gao et al. (2009), [1], Asymptotic Behaviors of Tail Density for Sum of Correlated Lognormal Variables. International Journal of Mathematics and Mathematical Sciences.
• Daniel Dufresne (2009), [2], SUMS OF LOGNORMALS, Centre for Actuarial Studies, University of Melbourne.

See also

• Normal distribution
• Geometric mean
• Geometric standard deviation
• Error function
• Log-distance path loss model
• Slow fading
• Stochastic volatility